We equip \(\mathfrak {sp}(2m,\mathbb {R})\)  with the Frobenius inner product \(\langle X,Y\rangle =\operatorname{tr}[X^\top Y]\) , which is not \(\operatorname{Ad}\) -invariant. In fact, there is no \(\operatorname{Ad}\) -invariant inner product on \(\mathfrak {sp}(2m,\mathbb {R})\)  [1]}.
As a vector space, \(\mathfrak {sp}(2m,\mathbb {R})\)  is isomorphic to the space of symmetric matrices \(\mathrm {Sym}(2m,\mathbb {R})={X\in \\\mathbb {R}^{2m\times 2m}X=X^\top }\)  via the map \(X \mapsto S=\Omega X\)  and its inverse \(S\mapsto X =-\Omega S\) .
In fact, this isomorphism is isometric w.r.t. the Frobenius inner product.
